CAM Colloquium-- Maria Cameron, Dept. of Math/Dept. of Computer Science, University of Maryland


655 Rhodes Hall


Title: Diffusion Maps for Solving PDEs in Moderately High Dimensions arising in Modeling Rare Events

Abstract: Modeling rare events such as conformal changes in molecules creates the need to solve high dimensional partial differential equations (PDEs). Traditional PDE solvers such as finite difference, finite element, or spectral methods are not suitable for such problems due to the need to mesh the ambient space and the curse of dimensionality. Novel numerical methods based on ideas borrowed from data science and machine learning are capable of overcoming this obstacle.

In this talk, I will show how diffusion maps (Coifman and Lafon, 2006) can be adapted for solving the backward Kolmogorov PDE arising in the context of rare event quantification. This approach does not require meshing the space, and its cost depends on the intrinsic dimensionality of the system. An approximation to the desired second-order differential operator is constructed by means of choosing an appropriate diffusion kernel and its renormalizations. A four-dimensional example concerned with transitions between metastable configurations of alanine dipeptide described in four dihedral angles will be demonstrated. Error estimates will be discussed.

Bio: I received my Ph.D. in Applied Mathematics from UC Berkeley in 2007. Next, I was a Courant Instructor at NYU. Since August 2010, I have been a professor in the Department of Mathematics of the University of Maryland, College Park. My research interests lie in applied and computational mathematics and are quite broad. My recent work includes applications to molecular dynamics, complex chemical reaction systems, nonlinear oscillators, and polymer networks. My recent contributions to scientific computing include methods based on optimal control, diffusion maps, and machine learning.